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On strong provability predicates and the associated modal logics

Published online by Cambridge University Press:  12 March 2014

Konstantin N. Ignatiev
Konstantin N. Ignatiev*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, E-mail: [email protected]
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Abstract

PA is Peano Arithmetic. Pr(x) is the usual Σ1,-formula representing provability in PA. A strong provability predicate is a formula which has the same properties as Pr(·) but is not Σ1. An example: Q is ω-provable if PA + ¬Q is ω-inconsistent (Boolos [4]). In [5] Dzhaparidze introduced a joint provability logic for iterated ω-provability and obtained its arithmetical completeness.

In this paper we prove some further modal properties of Dzhaparidze's logic, e.g., the fixed point property and the Craig interpolation lemma. We also consider other examples of the strong provability predicates and their applications.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 249 - 290
Copyright
Copyright © Association for Symbolic Logic 1993

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References

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